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Announcements

Announcements. Homework: Supplemental Problems 2 nd Project is due at the final exam which is 4:00pm Tuesday May 6. A list of potential projects is posted . Numbers 3 & 6 have already been taken.

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Announcements

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  1. Announcements • Homework: Supplemental Problems • 2nd Project is due at the final exam which is 4:00pm Tuesday May 6. A list of potential projects is posted. Numbers 3 & 6 have already been taken. • Final Exam will be another hour exam covering the material we cover since the last exam (relativity and black holes). It will be the second thing during the final exam period (project presentation first)

  2. Black Holes! Black holes are places where space-time is warped beyond the breaking point.

  3. The physics of black holes is Einstein’s General Theory of Relativity For the full form, see the Einstein’s Field Equation link on the class website

  4. The Schwarzschild Metric Karl Schwarzschild worked out Einstein’s equations of general relativity for a non-rotating point mass while serving as an artillery officer in the German army in WWI. His “metric” describes space-time near the object. The Schwarzschild radius is given by

  5. Consequences of the Schwarzschild Metric Time Dilation: a clock at a distance R from the singularity will appear to run slow as compared to a clock at infinity Length Stretching: a meter stick placed at a distance R from the singularity will appear longer as measured by an observer at infinity Watch Spaghettification video

  6. Examples Determine the Schwarzschild radius of the Sun. By what factor is time slowed for a clock placed at twice the Schwarzschild radius from the singularity of a black hole? By what factor is a 1.0 m long meter stick appear lengthened to an observer far away if it placed at twice the Schwarzschild radius from the singularity of a black hole?

  7. Example Solution From Appendix 2 of the textbook Msun = 1.9891 x 1030 kg The clock near the black hole runs about 70.7% as fast as the far away clock (or 29.3% slower) To a far away observer, the 1.00 m stick would appear to be 1.41 m long

  8. Rotating Black Holes

  9. A Kerr black hole has two event horizons

  10. The two even horizons are related to the angular momentum L is the angular momentum of the black hole and q is the angle from the rotation axis of the black hole

  11. Example Determine the equatorial radius of the two event horizons of a 5.0 solar mass black hole that is rotating at 20 times per second. Assume a spherical mass with a radius of 1000.0 m to determine the angular momentum First, find the angular momentum of the black hole Next, find “a”

  12. Example Solution Now plug the mass and “a” into the equations for the two radii At the equator q = 90° so cosq = 0 thus

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